electromagnetism

Moving electrons are always surrounded by closed magnetic lines of force lying in a plane perpendicular to their motion. That is, an electric current generates a magneticfield which surrounds the conductor carrying the current and is at right angles to the direction of current flow. The field disappears when the current is shut off, because air has no magnetic retentivity.
The strength of the magnetic field set up by an electromagnet depends on the amplitude of the current, the number of turns in the coil, the shape of the coil, the magnetic properties of the material surrounding the coil, and the lenght of the magnetic path. Nonferrousmaterials are in general the same air in their magnetic properties.

The convergence occurs partly because of Maxwell's Equations which shows how they interelate for a fixed frame of reference.

However, that they are the same thing is most elegantly described by Special Relativity; which shows how simple relative motion of a frame of reference can convert an electrostatic field into a magnetic field (and vice versa). In fact they form what is known as a four vector.

Due to Relativity some observers can correctly state that an effect is caused by an electrostatic field, whilst others will think the same effect is magnetic in origin, but all agree on what actually transpired. And thus this shows that they are the self same thing.